Two numbers have a difference of 10. The sum of the larger one and two thirds of the smaller number is 16. What are these two numbers?

Question

Two numbers have a difference of 10. The sum of the larger one and two thirds of the smaller number is 16. What are these two numbers?

Answer 1

Let's call the larger number y and the smaller number x.

 

When we convert these words into expressions/equations, we get the following:

larger-smaller=10

y-x=10

 

sum of the larger and 2/3 smaller =y+(2/3)x=16

 

Solve the system of equations using substitution or elimination.

Substitution seems to be the easier method in this case.

 

y-x=10

y-10-x=0

y-10=x

 

y+(2/3)x=16

y+(2/3)(y-10)=16

y+2y/3-20/3=16

y+2y/3=16+20/3

5y/3=68/3

y=(68/3)/(5/3)=(68/3)(3/5)=68/5=larger number

 

y-x=10

68/5-x=10

68/5-10=x

(68-5*10)/5=18/5=smaller number

 

The two numbers are 18/5 and 68/5.

Answer 2

x - y = 10  and

x + 2y/3 = 16

Subtract

 - 5y/3 = - 6

y = 18/5  => x = 16 - 2/3 (18/5)  = 204/15

==>So x , y = 54/15 , 204/15  <==

 

Check  

(204 - 54)/15 = 150/15 = 10